What is the solution for (d^2y)/(dx^2)-6(dy)/(dx)+13y=0 ?

The solution for (d^2y)/(dx^2)-6(dy)/(dx)+13y=0 is y=e^{3x}(c_{1}cos(2x)+c_{2}sin(2x))

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(d^2y)/(dx^2)-6(dy)/(dx)+13y=0

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 13 y = 0

y = e^{3 x} (c_{1} cos (2 x) + c_{2} sin (2 x))

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 13 y = 0

Solve linear ODE:

y = e^{3 x} (c_{1} cos (2 x) + c_{2} sin (2 x))

y = e^{3 x} (c_{1} cos (2 x) + c_{2} sin (2 x))

2 y^{^{''}} - 5 y^{^{'}} = 0

6 y^{^{''}} - 5 y^{^{'}} + y = 0, y (0) = - 1, y^{^{'}} (0) = 1

(D^{2} + 9 D + 18) y = 0

y^{^{'''}} - y^{^{''}} - 5 y^{^{'}} - 3 y = 0

x^{^{''}} + 10 x^{^{'}} + 16 x = 0

What is the solution for (d^2y)/(dx^2)-6(dy)/(dx)+13y=0 ?
The solution for (d^2y)/(dx^2)-6(dy)/(dx)+13y=0 is y=e^{3x}(c_{1}cos(2x)+c_{2}sin(2x))